# Simplicial set represented by an (unordered) set

Let $$X$$ be a (finite if you want) set and form the simplicial set $$F^{\bullet}(X)$$ with $$F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X)$$ where the right hand side denotes arbitrary maps of sets (of course it wouldn't make sense to say order preserving as $$X$$ doesn't come with an order).

I'm wondering about a description of $$F^{\bullet}(X)$$. For example if $$X = \{0,1\}$$ then there are 2 0-simplices, may as well call them $$ and $$ and 2 1-simplices $$[0, 1]$$ and $$[1,0]$$ glued together to form a copy of $$S^1$$.

Edit: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.

Is there an analogous description when $$X = \{0, 1, 2\}$$?

A closely related question is whether there's a right adjoint to the forgetful functor from the simplex category $$\Delta$$ (finite ordered sets) to, say, finite (unordered) sets -- and if so what is it.

Example where such simplicial sets arise: given a map of topological spaces $$f: X \to Y$$ we can always form a simplicial object $$\mathcal{S}^{\bullet}(f)$$ with $$\mathcal{S}^{n} = \prod\nolimits_{X}^{n} = \underbrace{X \times_{Y} \cdots \times_{Y} X}_{n\text{ times }}$$ with face and degeneracy maps given by projections and diagonals respectively. Taking connected components gives a simplicial set.

When $$Y$$ is the union $$\bigcup_{i=1}^{N} H_{i}$$ of the coordinate hyperplanes in $$\mathbb{C}^{N}$$ and $$f: X=\coprod_{i=1}^{N} H_{i} \to \bigcup_{i=1}^{N} H_{i}=Y$$ is the obvious map, I believe the simplicial set we get is $$F^{\bullet}(\{1, \dots, n\})$$.

## 1 Answer

You are overlooking some nondegenerate simplices. For example, when $$X={0,1}$$ there are the $$2$$-cells $$[0,1,0]$$ and $$[1,0,1]$$. In fact, the thing you call $$F^\bullet(X)$$ is infinite dimensional if $$X$$ has more than one element.

It is contractible whenever $$X$$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.

If $$X=G$$ has a group structure then $$F^\bullet(G)$$ is often called $$EG$$; it is a contractible space with free $$G$$-action.

• Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^\bullet(X)$ as the nerve of the category with objects the points $x \in X$ and with a unique morphism $x \to y$ for every 2 points $x, y \in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x \in X$ with its identity morphism would give an equivalence of categories ... – cgodfrey Apr 29 at 2:26
• Yes. That's what I meant in my second paragraph! – Tom Goodwillie Apr 29 at 20:55